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Math 512, Fall 14 Combinatorial Set Theory Week 1 MATH 512, FALL 14 COMBINATORIAL SET THEORY WEEK 1 Let κ be a regular (i.e. cf(κ) = κ) uncountable cardinal. Definition 1. A set C ⊂ κ is closed and unbounded (club) in κ if • for all α < κ, there is β 2 C n α, and • for all increasing sequences hαi j i < τi of ordinals in C for some τ < κ, supi<τ αi 2 C. Some examples: κ n α, for all α < κ; the set of limit ordinals less than κ. Lemma 2. Suppose that C; D are clubs in κ. Then so is C \ D. Proof. To show closure, suppose that τ < κ and hαi j i < τi is an increasing sequence of ordinals in C \ D, and let α = supξ<τ αξ. Since C is closed, we have that α 2 C. Since D is closed, we have that α 2 D. To show unboundedness, fix α < κ. Let α0 > α be a point in C. Then let β0 > α0 be a point in D. Continue inductively, to build increasing sequences hαn j n < !i of points in C and hβn j n < !i of points in D, such that for all n, αn < βn < αn+1. Then β := supn αn = supn βn 2 C \ D, and β > α. Definition 3. A set S ⊂ κ is stationary if for all clubs C, S \ C 6= ; κ Some examples: every club set, E! := fα < κ j cf(α) = !g. Also note that if S is stationary and C is a club, then S \ C is stationary. Definition 4. Let hAξ j ξ < κi be subsets of κ. The diagonal intersection is T defined to be 4ξ<κAξ := fβ < κ j β 2 ξ<β Aξg. A filter is called normal if it is closed under diagonal intersections. Proposition 5. T (1) If hCξ j ξ < τi are clubs in κ for some τ < κ, then ξ<τ Cξ is also a club. (2) If hCξ j ξ < κi are clubs in κ, then 4ξ<τ Cξ is a club. The club filter on κ is the collection of all subsets of κ containing a club. If follows by the above that the club filter is a normal κ-complete filter on κ, and it contains all complements of bounded sets. Theorem 6. (Fodor) Suppose that S ⊂ κ is stationary and f : S ! κ is a regressive function, i.e. f(α) < α for all α 2 S. Then there is a stationary T ⊂ S, such that f is constant on T . Proof. Otherwise, for all γ < κ, f −1(γ) is nonstationary, i.e. there is a club −1 Cγ with Cγ \ f (γ) = ;. Let C = 4γ<κCγ, and let α 2 C \ S. Set 1 2 MATH 512, FALL 14 COMBINATORIAL SET THEORY WEEK 1 γ := f(α). Since f is regressive, γ < α, and so by the definition of diagonal −1 intersection, α 2 Cγ. But α 2 f (γ). Contradiction. The conclusion of this theorem is actually a necessary and sufficient con- dition for normality. An application of Fodor's theorem is the following fact: Fact (Solovay): Every stationary subset S of κ can be partitioned into κ many disjoint stationary subsets. (for the proof, see Chapter 8 of Jech) Definition 7. A cardinal κ is inaccessible if it is regular and strong limit (i.e. τ < κ ! 2τ < κ). Proposition 8. Suppose κ is inaccessible. Then the set of cardinals below κ is club. Definition 9. An inaccessible cardinal κ is Mahlo if the set of regular car- dinals below κ is stationary. So far we have defined clubs and stationary subsets of a cardinal. For an ordinal α, c ⊂ α is a club in α is it is closed and unbounded in α, and s ⊂ α is stationary in α is it meets every club in α. For a set B ⊂ β, lim(B) will denote the limit points of B, i.e. lim(B) := fα < β j B \ α is unboundedg. Note that if B is unbounded, then lim(B) is a club. Also, for any club C, lim(C) ⊂ C. Definition 10. Let S ⊂ κ be stationary. S reflects if for some α < κ, S \ α is stationary in α. Definition 11. For a stationary set T , Refl(T ) denotes the statement that every stationary subset of T reflects. κ For example, for any uncountable regular κ, E! reflects. Proposition 12. Let κ be any cardinal (possibly singular), and let T be a κ+ stationary subset of Eκ . Then T does not reflect. Proof. Let α < κ+ be any point. Let C ⊂ α be club in α with o:t:(C) = cf(α) ≤ κ. Then lim(C) is a club subset of α, disjoint from T . + Definition 13. κ asserts the existence of a sequence hCα j α < κ i, such that for every α, • Cα is club in α with o:t:(Cα) ≤ κ; • if β 2 lim(Cα), then Cα \ β = Cβ. + Lemma 14. κ implies :Refl(S) for every stationary S ⊂ κ . + Proof. Suppose that hCα j α < κ i is a square sequence and S is a stationary + subset of κ . Let F (α) := o:t:(Cα). By Fodor, there is a stationary subset T ⊂ S, such that F is constant on T . I.e. for some δ, for all α 2 T , o:t:(Cα) = δ. We claim that T does not reflect. For otherwise, if T \ α is MATH 512, FALL 14 COMBINATORIAL SET THEORY WEEK 1 3 stationary in α, then T \ lim(Cα) is also stationary in α. Let β < γ be two points in T \ lim(Cα). Then since Cα \ γ = Cγ, we have that β 2 lim(Cγ), and so Cγ \ β = Cβ. It follows that F (β) = o:t:(Cβ) < o:t:(Cγ) = F (γ). Contradiction with β; γ 2 T . Next we give some weakenings of square: + Definition 15. κ,λ asserts the existence of a sequence hCα j α < κ i, such that for every α, • 1 ≤ jCαj ≤ λ, • for every α, for every C 2 Cα, C is club in α with o:t:(C) ≤ κ; • for every α, for every C 2 Cα, for every β 2 lim(C), we have C \β 2 Cβ. ∗ The principle weak square is κ := κ,κ. We have that for any λ < κ, ∗ κ ! κ,λ ! κ. <κ ∗ Lemma 16. Suppose that κ = κ. Then κ holds. + Proof. For every limit α < κ with cf(α) < κ, let Cα := fC ⊂ α j C is a club, jCj < κg, i.e. all club subsets of α of size less than κ. Since <κ κ = κ, we have that jCαj = κ. + If κ is regular, for every limit α < κ with cf(α) = κ, let Cα be any club in α of order type κ. Set Cα = fCαg. Suppose that C 2 Cα and β 2 lim(C). Then since jCj ≤ κ, we have that cf(β) < κ and C \ β is a club subset of β of size less than κ. So, by definition, C \ β 2 Cβ. Note that the above implies that weak square holds for all inaccessible κ. Also, under GCH, weak square will hold for every regular κ. So, we will be ∗ most interested in κ when κ is singular. In general square principles are a \incompactness" type properties: a property that a structure lacks, but all of its substructures of smaller cardi- nality have. This is examplified in the following lemma: + Lemma 17. Suppose that hCα j α < κ i is a κ,λ sequence, for some 1 ≤ λ ≤ κ. Then there is no club C ⊂ κ+, such that for all α 2 lim(C), C \ α 2 Cα. Proof. If C is a club in κ+, let α 2 lim(C) be such that o:t:(C \ α) > κ. + We can always find such an α, since o:t:(C) = κ . But for every E 2 Cα, o:t:(E) ≤ κ, so C \ α2 = Cα. Definition 18. A tree (T; <) is a partially ordered set, such that for every x 2 T , the set of predecessors of x, pred(x) := fy 2 T j y < xg is well ordered by <. We set: • for x 2 T , level(x) := o:t:(pred(x)), 4 MATH 512, FALL 14 COMBINATORIAL SET THEORY WEEK 1 • the height of the tree, ht(T ) := supflevel(x) + 1 j x 2 T g, • for α < ht(T ), the α-th level of T is Tα := fx 2 T j level(x) = αg. We also say that b ⊂ T is a branch, if b is a maximal linearly ordered subset of T . Note that if b is a branch, then for every level α, jb \ Tαj ≤ 1, and if β < α, then by maximality b \ Tα 6= ; implies that b \ Tβ 6= ;. We say that b is unbounded (or cofinal) if for all α < ht(T ), b \ Tα 6= ;. Definition 19. The tree property holds at κ, for a regular cardinal κ, if every tree of height κ and levels of size less than κ, has an unbounded branch. We denote this by TPκ. Below we list some facts about the tree property: (1) (K¨onig)TP holds at !. (2) (Aronszajn) TP fails at !1. (3) TP can hold at !2 (and above), assuming some large cardinals. ∗ + (4) κ implies that the tree property fails at κ . Definition 20. An inaccessible cardinal κ is weakly compact if it satisfies the tree property. There are several equivalent definitions of a weakly compact.
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